Conventional Partial and Complete Solutions of the Fundamental Equations of Fluid Mechanics in the Problem of Periodic Internal Waves with Accompanying Ligaments Generation

Yuli D Chashechkin

doi:10.3390/math9060586

Abstract

The problem of generating beams of periodic internal waves in a viscous, exponentially stratified fluid by a band oscillating along an inclined plane is considered by the methods of the theory of singular perturbations in the linear and weakly nonlinear approximations. The complete solution to the linear problem, which satisfies the boundary conditions on the emitting surface, is constructed taking into account the previously proposed classification of flow structural components described by complete solutions of the linearized system of fundamental equations without involving additional force or mass sources. Analyses includes all components satisfying the dispersion relation that are periodic waves and thin accompanying ligaments, the transverse scale of which is determined by the kinematic viscosity and the buoyancy frequency. Ligaments are located both near the emitting surface and in the bulk of the liquid in the form of wave beam envelopes. Calculations show that in a nonlinear description of all components, both waves and ligaments interact directly with each other in all combinations: waves-waves, waves-ligaments, and ligaments-ligaments. Direct interactions of the components that generate new harmonics of internal waves occur despite the differences in their scales. Additionally, the problem of generating internal waves by a rapidly bi-harmonically oscillating vertical band is considered. If the difference in the frequencies of the spectral components of the band movement is less than the buoyancy frequency, the nonlinear interacting ligaments generate periodic waves as well. The estimates made show that the amplitudes of such waves are large enough to be observed under laboratory conditions.

Highlights

Stable interest in the systematic scientific study of internal gravity waves (IGW), which are hidden relatively slow periodic flows inside the stratified atmosphere and ocean, was only formed in the second half of the last century in connection with the general development of Earth sciences
It is more natural to consider the complex wavenumber k = k1 + ik2, the imaginary part of which k2 characterizes the spatial attenuation and restructuring of the wave beam during propagation or reflection from a solid plane [20,21]. Another independent approach was developed for the description of periodic flows in a viscous stratified fluid based on the construction of complete solutions of the linearized system of governing equations
In the low-viscosity approximation, the theory of singular perturbations [23] gives room to calculate complete solutions for internal wave equations taking into account the condition of compatibility

Summary

Introduction

Stable interest in the systematic scientific study of internal gravity waves (IGW), which are hidden relatively slow periodic flows inside the stratified atmosphere and ocean, was only formed in the second half of the last century in connection with the general development of Earth sciences. It is more natural to consider the complex wavenumber k = k1 + ik , the imaginary part of which k2 characterizes the spatial attenuation and restructuring of the wave beam during propagation or reflection from a solid plane [20,21] Another independent approach was developed for the description of periodic flows in a viscous stratified fluid based on the construction of complete solutions of the linearized system of governing equations. In the low-viscosity approximation, the theory of singular perturbations [23] gives room to calculate complete solutions for internal wave equations taking into account the condition of compatibility. The band periodically oscillating with frequency ω generates viscous exponentially stratified fluid with vertical density profile ρ0 = ρ00 exp(−z/Λ) and constant buoyancy frequency N periodical perturbations containing internal wave beams and ligaments. Taking into account these properties, the expression r for the stream in the region ξ>a/2

N cos θ

Non-Linear Generation of Flows by a Band Oscillating along a Vertical Plate

Discussion of the Results